A similar theory for topological systems has been elusive due to additional topological terms that enter into the conservation law (REF ), see (REF ) below. Local existence and uniqueness of solutions in class (REF ) for a large \(m\in \mathbb {N}\)  was established in [1]} by a direct and technical application of fractional energy estimates. This result comes with a rough continuation criterion in terms of \(u,\rho \in C^2\) . With more effort the technique gives \(u,\rho \in C^\gamma \) , where \(\gamma = 1+\varepsilon \)  for \(\alpha \le 1\) , and \(\gamma = \alpha +\varepsilon \)  for \(\alpha >1\) . In either case such a criterion is less effective from the application standpoint as the typical fractional parabolic regularity theory, for example, provides Schauder estimates in \(C^{1+\delta }\)  for an indeterminate small parameter \(\delta >0\) , see [2]}, [3]}, [4]} and references therein.
